2D Mensuration • Topic 2 of 3

Circle, Sector & Segment

A full circle has area πr² and circumference 2πr. A sector is a "pizza slice" subtending angle θ at the centre: its area is the fraction θ/360 of the whole circle, (θ/360)πr², and its arc length is the same fraction of the circumference, (θ/360)·2πr. The perimeter of a sector is that arc plus the two radii: arc + 2r. A segment is the region between a chord and its arc; the minor segment equals the sector minus the triangle formed by the two radii and the chord, so its area is (θ/360)πr² − ½r²·sinθ. CAT loves the clean angles — 60° gives a sixth of the circle, 90° a quarter, 120° a third — so recognise the fraction instantly. Keep answers in terms of π unless a numeric value is demanded; substituting 22/7 or 3.14 too early is the main source of error and wasted time.

✅ Solved examples

1. Find the area and circumference of a circle of radius 7 cm (use π = 22/7).
Area = πr² = (22/7)(49) = 154 cm². Circumference = 2πr = 2(22/7)(7) = 44 cm.
2. A sector of a circle of radius 6 cm subtends 60° at the centre. Find its area.
(θ/360)πr² = (60/360)π(36) = (1/6)(36π) = 6π ≈ 18.85 cm².
3. Find the arc length of a 90° sector in a circle of radius 14 cm (π = 22/7).
Arc = (90/360)·2πr = ¼ × 2 × (22/7) × 14 = ¼ × 88 = 22 cm.
4. A chord subtends 90° at the centre of a circle of radius 10 cm. Find the area of the minor segment.
Sector = ¼π(100) = 25π. Triangle = ½r²·sin90° = ½(100)(1) = 50. Segment = 25π − 50 ≈ 28.54 cm².

✏️ Practice — try these, take hints as needed

1. Area of a circle of radius 10 cm (in terms of π)?
πr².
r² = 100.
Leave π in.
100π cm² (≈ 314.16)
2. Circumference of a circle of diameter 28 cm (π = 22/7)?
C = πd.
(22/7) × 28.
22 × 4.
88 cm
3. Area of a 120° sector in a circle of radius 9 cm (in terms of π)?
Fraction = 120/360 = 1/3.
(1/3)π(81).
81/3 × π.
27π cm² (≈ 84.82)
4. Arc length of a 45° sector, radius 8 cm (in terms of π)?
(45/360)·2πr.
⅛ × 16π.
16π/8.
2π cm (≈ 6.28)
5. Minor segment area for a 60° chord in a circle of radius 6 cm?
Sector = (1/6)π(36) = 6π.
Triangle = ½(36)sin60° = 9√3.
Subtract.
6π − 9√3 ≈ 3.26 cm²

📝 Topic test — 8 questions

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